A mountain pass theorem for minimal hypersurfaces with fixed boundary

TL;DR

This paper develops min-max methods to prove the existence of a third embedded minimal hypersurface with fixed boundary, extending techniques to a discrete setting and building on prior stable hypersurface results.

Contribution

It introduces a new min-max approach adapted to Almgren-Pitts theory to establish the existence of an additional minimal hypersurface with fixed boundary.

Findings

Proves existence of a third minimal hypersurface with fixed boundary.

Adapts min-max methods to a discrete Almgren-Pitts setting.

Extends previous results on stable minimal hypersurfaces.

Abstract

In this work, we prove the existence of a third embedded minimal hypersurface spanning a closed submanifold $\gamma$ contained in the boundary of a compact Riemannian manifold with convex boundary, when it is known a priori the existence of two strictly stable minimal hypersurfaces that bound $\gamma$. In order to do so, we develop min-max methods similar to those of De Lellis and Ramic, references in the paper, adapted to the discrete setting of Almgren and Pitts.